Calculating the algebraic entropy of mappings with unconfined singularities

TL;DR

This paper introduces a method to compute the algebraic entropy of mappings with unconfined singularities, enabling analysis of their complexity and integrability properties.

Contribution

It extends Halburd's method to mappings with unconfined singularities, allowing exact calculation of dynamical degrees and algebraic entropy.

Findings

Method successfully computes dynamical degree for nonintegrable mappings

The approach yields zero entropy for linearisable mappings with unconfined singularities

Examples demonstrate the method's effectiveness in various cases

Abstract

We present a method for calculating the dynamical degree of a mapping with unconfined singularities. It is based on a method introduced by Halburd for the computation of the growth of the iterates of a rational mapping with confined singularities. In particular, we show through several examples how simple calculations, based on the singularity patterns of the mapping, allow one to obtain the exact value of the dynamical degree for nonintegrable mappings that do not possess the singularity confinement property. We also study linearisable mappings with unconfined singularities to show that in this case our method indeed yields zero algebraic entropy.